Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q22

Background

All theorems, corollaries, and definitions listed in the book's order:

Theorems and Corollaries in Elementary Number Theory (Ch 1 - 3)

All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 3)

RanblogRan

I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.

Question

Show that $13$ is the largest prime that can divide two successive integers of the form $n^{2} + 3$.

Solution

Let $p$ be a prime a such that $p \mid k^{2} + 3$ and $p \mid (k + 1)^{2} + 3$ for some integer $k$.